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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math.transform;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import java.io.Serializable;<a name="line.19"></a>
<FONT color="green">020</FONT>    import java.lang.reflect.Array;<a name="line.20"></a>
<FONT color="green">021</FONT>    <a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math.FunctionEvaluationException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math.MathRuntimeException;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math.analysis.UnivariateRealFunction;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math.complex.Complex;<a name="line.25"></a>
<FONT color="green">026</FONT>    <a name="line.26"></a>
<FONT color="green">027</FONT>    /**<a name="line.27"></a>
<FONT color="green">028</FONT>     * Implements the &lt;a href="http://mathworld.wolfram.com/FastFourierTransform.html"&gt;<a name="line.28"></a>
<FONT color="green">029</FONT>     * Fast Fourier Transform&lt;/a&gt; for transformation of one-dimensional data sets.<a name="line.29"></a>
<FONT color="green">030</FONT>     * For reference, see &lt;b&gt;Applied Numerical Linear Algebra&lt;/b&gt;, ISBN 0898713897,<a name="line.30"></a>
<FONT color="green">031</FONT>     * chapter 6.<a name="line.31"></a>
<FONT color="green">032</FONT>     * &lt;p&gt;<a name="line.32"></a>
<FONT color="green">033</FONT>     * There are several conventions for the definition of FFT and inverse FFT,<a name="line.33"></a>
<FONT color="green">034</FONT>     * mainly on different coefficient and exponent. Here the equations are listed<a name="line.34"></a>
<FONT color="green">035</FONT>     * in the comments of the corresponding methods.&lt;/p&gt;<a name="line.35"></a>
<FONT color="green">036</FONT>     * &lt;p&gt;<a name="line.36"></a>
<FONT color="green">037</FONT>     * We require the length of data set to be power of 2, this greatly simplifies<a name="line.37"></a>
<FONT color="green">038</FONT>     * and speeds up the code. Users can pad the data with zeros to meet this<a name="line.38"></a>
<FONT color="green">039</FONT>     * requirement. There are other flavors of FFT, for reference, see S. Winograd,<a name="line.39"></a>
<FONT color="green">040</FONT>     * &lt;i&gt;On computing the discrete Fourier transform&lt;/i&gt;, Mathematics of Computation,<a name="line.40"></a>
<FONT color="green">041</FONT>     * 32 (1978), 175 - 199.&lt;/p&gt;<a name="line.41"></a>
<FONT color="green">042</FONT>     *<a name="line.42"></a>
<FONT color="green">043</FONT>     * @version $Revision: 790243 $ $Date: 2009-07-01 12:03:28 -0400 (Wed, 01 Jul 2009) $<a name="line.43"></a>
<FONT color="green">044</FONT>     * @since 1.2<a name="line.44"></a>
<FONT color="green">045</FONT>     */<a name="line.45"></a>
<FONT color="green">046</FONT>    public class FastFourierTransformer implements Serializable {<a name="line.46"></a>
<FONT color="green">047</FONT>    <a name="line.47"></a>
<FONT color="green">048</FONT>        /** Serializable version identifier. */<a name="line.48"></a>
<FONT color="green">049</FONT>        static final long serialVersionUID = 5138259215438106000L;<a name="line.49"></a>
<FONT color="green">050</FONT>    <a name="line.50"></a>
<FONT color="green">051</FONT>        /** roots of unity */<a name="line.51"></a>
<FONT color="green">052</FONT>        private RootsOfUnity roots = new RootsOfUnity();<a name="line.52"></a>
<FONT color="green">053</FONT>    <a name="line.53"></a>
<FONT color="green">054</FONT>        /**<a name="line.54"></a>
<FONT color="green">055</FONT>         * Construct a default transformer.<a name="line.55"></a>
<FONT color="green">056</FONT>         */<a name="line.56"></a>
<FONT color="green">057</FONT>        public FastFourierTransformer() {<a name="line.57"></a>
<FONT color="green">058</FONT>            super();<a name="line.58"></a>
<FONT color="green">059</FONT>        }<a name="line.59"></a>
<FONT color="green">060</FONT>    <a name="line.60"></a>
<FONT color="green">061</FONT>        /**<a name="line.61"></a>
<FONT color="green">062</FONT>         * Transform the given real data set.<a name="line.62"></a>
<FONT color="green">063</FONT>         * &lt;p&gt;<a name="line.63"></a>
<FONT color="green">064</FONT>         * The formula is $ y_n = \Sigma_{k=0}^{N-1} e^{-2 \pi i nk/N} x_k $<a name="line.64"></a>
<FONT color="green">065</FONT>         * &lt;/p&gt;<a name="line.65"></a>
<FONT color="green">066</FONT>         * <a name="line.66"></a>
<FONT color="green">067</FONT>         * @param f the real data array to be transformed<a name="line.67"></a>
<FONT color="green">068</FONT>         * @return the complex transformed array<a name="line.68"></a>
<FONT color="green">069</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.69"></a>
<FONT color="green">070</FONT>         */<a name="line.70"></a>
<FONT color="green">071</FONT>        public Complex[] transform(double f[])<a name="line.71"></a>
<FONT color="green">072</FONT>            throws IllegalArgumentException {<a name="line.72"></a>
<FONT color="green">073</FONT>            return fft(f, false);<a name="line.73"></a>
<FONT color="green">074</FONT>        }<a name="line.74"></a>
<FONT color="green">075</FONT>    <a name="line.75"></a>
<FONT color="green">076</FONT>        /**<a name="line.76"></a>
<FONT color="green">077</FONT>         * Transform the given real function, sampled on the given interval.<a name="line.77"></a>
<FONT color="green">078</FONT>         * &lt;p&gt;<a name="line.78"></a>
<FONT color="green">079</FONT>         * The formula is $ y_n = \Sigma_{k=0}^{N-1} e^{-2 \pi i nk/N} x_k $<a name="line.79"></a>
<FONT color="green">080</FONT>         * &lt;/p&gt;<a name="line.80"></a>
<FONT color="green">081</FONT>         * <a name="line.81"></a>
<FONT color="green">082</FONT>         * @param f the function to be sampled and transformed<a name="line.82"></a>
<FONT color="green">083</FONT>         * @param min the lower bound for the interval<a name="line.83"></a>
<FONT color="green">084</FONT>         * @param max the upper bound for the interval<a name="line.84"></a>
<FONT color="green">085</FONT>         * @param n the number of sample points<a name="line.85"></a>
<FONT color="green">086</FONT>         * @return the complex transformed array<a name="line.86"></a>
<FONT color="green">087</FONT>         * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.87"></a>
<FONT color="green">088</FONT>         * at some point<a name="line.88"></a>
<FONT color="green">089</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.89"></a>
<FONT color="green">090</FONT>         */<a name="line.90"></a>
<FONT color="green">091</FONT>        public Complex[] transform(UnivariateRealFunction f,<a name="line.91"></a>
<FONT color="green">092</FONT>                                   double min, double max, int n)<a name="line.92"></a>
<FONT color="green">093</FONT>            throws FunctionEvaluationException, IllegalArgumentException {<a name="line.93"></a>
<FONT color="green">094</FONT>            double data[] = sample(f, min, max, n);<a name="line.94"></a>
<FONT color="green">095</FONT>            return fft(data, false);<a name="line.95"></a>
<FONT color="green">096</FONT>        }<a name="line.96"></a>
<FONT color="green">097</FONT>    <a name="line.97"></a>
<FONT color="green">098</FONT>        /**<a name="line.98"></a>
<FONT color="green">099</FONT>         * Transform the given complex data set.<a name="line.99"></a>
<FONT color="green">100</FONT>         * &lt;p&gt;<a name="line.100"></a>
<FONT color="green">101</FONT>         * The formula is $ y_n = \Sigma_{k=0}^{N-1} e^{-2 \pi i nk/N} x_k $<a name="line.101"></a>
<FONT color="green">102</FONT>         * &lt;/p&gt;<a name="line.102"></a>
<FONT color="green">103</FONT>         * <a name="line.103"></a>
<FONT color="green">104</FONT>         * @param f the complex data array to be transformed<a name="line.104"></a>
<FONT color="green">105</FONT>         * @return the complex transformed array<a name="line.105"></a>
<FONT color="green">106</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.106"></a>
<FONT color="green">107</FONT>         */<a name="line.107"></a>
<FONT color="green">108</FONT>        public Complex[] transform(Complex f[])<a name="line.108"></a>
<FONT color="green">109</FONT>            throws IllegalArgumentException {<a name="line.109"></a>
<FONT color="green">110</FONT>            roots.computeOmega(f.length);<a name="line.110"></a>
<FONT color="green">111</FONT>            return fft(f);<a name="line.111"></a>
<FONT color="green">112</FONT>        }<a name="line.112"></a>
<FONT color="green">113</FONT>    <a name="line.113"></a>
<FONT color="green">114</FONT>        /**<a name="line.114"></a>
<FONT color="green">115</FONT>         * Transform the given real data set.<a name="line.115"></a>
<FONT color="green">116</FONT>         * &lt;p&gt;<a name="line.116"></a>
<FONT color="green">117</FONT>         * The formula is $y_n = (1/\sqrt{N}) \Sigma_{k=0}^{N-1} e^{-2 \pi i nk/N} x_k$<a name="line.117"></a>
<FONT color="green">118</FONT>         * &lt;/p&gt;<a name="line.118"></a>
<FONT color="green">119</FONT>         * <a name="line.119"></a>
<FONT color="green">120</FONT>         * @param f the real data array to be transformed<a name="line.120"></a>
<FONT color="green">121</FONT>         * @return the complex transformed array<a name="line.121"></a>
<FONT color="green">122</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.122"></a>
<FONT color="green">123</FONT>         */<a name="line.123"></a>
<FONT color="green">124</FONT>        public Complex[] transform2(double f[])<a name="line.124"></a>
<FONT color="green">125</FONT>            throws IllegalArgumentException {<a name="line.125"></a>
<FONT color="green">126</FONT>    <a name="line.126"></a>
<FONT color="green">127</FONT>            double scaling_coefficient = 1.0 / Math.sqrt(f.length);<a name="line.127"></a>
<FONT color="green">128</FONT>            return scaleArray(fft(f, false), scaling_coefficient);<a name="line.128"></a>
<FONT color="green">129</FONT>        }<a name="line.129"></a>
<FONT color="green">130</FONT>    <a name="line.130"></a>
<FONT color="green">131</FONT>        /**<a name="line.131"></a>
<FONT color="green">132</FONT>         * Transform the given real function, sampled on the given interval.<a name="line.132"></a>
<FONT color="green">133</FONT>         * &lt;p&gt;<a name="line.133"></a>
<FONT color="green">134</FONT>         * The formula is $y_n = (1/\sqrt{N}) \Sigma_{k=0}^{N-1} e^{-2 \pi i nk/N} x_k$<a name="line.134"></a>
<FONT color="green">135</FONT>         * &lt;/p&gt;<a name="line.135"></a>
<FONT color="green">136</FONT>         * <a name="line.136"></a>
<FONT color="green">137</FONT>         * @param f the function to be sampled and transformed<a name="line.137"></a>
<FONT color="green">138</FONT>         * @param min the lower bound for the interval<a name="line.138"></a>
<FONT color="green">139</FONT>         * @param max the upper bound for the interval<a name="line.139"></a>
<FONT color="green">140</FONT>         * @param n the number of sample points<a name="line.140"></a>
<FONT color="green">141</FONT>         * @return the complex transformed array<a name="line.141"></a>
<FONT color="green">142</FONT>         * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.142"></a>
<FONT color="green">143</FONT>         * at some point<a name="line.143"></a>
<FONT color="green">144</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.144"></a>
<FONT color="green">145</FONT>         */<a name="line.145"></a>
<FONT color="green">146</FONT>        public Complex[] transform2(UnivariateRealFunction f,<a name="line.146"></a>
<FONT color="green">147</FONT>                                    double min, double max, int n)<a name="line.147"></a>
<FONT color="green">148</FONT>            throws FunctionEvaluationException, IllegalArgumentException {<a name="line.148"></a>
<FONT color="green">149</FONT>    <a name="line.149"></a>
<FONT color="green">150</FONT>            double data[] = sample(f, min, max, n);<a name="line.150"></a>
<FONT color="green">151</FONT>            double scaling_coefficient = 1.0 / Math.sqrt(n);<a name="line.151"></a>
<FONT color="green">152</FONT>            return scaleArray(fft(data, false), scaling_coefficient);<a name="line.152"></a>
<FONT color="green">153</FONT>        }<a name="line.153"></a>
<FONT color="green">154</FONT>    <a name="line.154"></a>
<FONT color="green">155</FONT>        /**<a name="line.155"></a>
<FONT color="green">156</FONT>         * Transform the given complex data set.<a name="line.156"></a>
<FONT color="green">157</FONT>         * &lt;p&gt;<a name="line.157"></a>
<FONT color="green">158</FONT>         * The formula is $y_n = (1/\sqrt{N}) \Sigma_{k=0}^{N-1} e^{-2 \pi i nk/N} x_k$<a name="line.158"></a>
<FONT color="green">159</FONT>         * &lt;/p&gt;<a name="line.159"></a>
<FONT color="green">160</FONT>         * <a name="line.160"></a>
<FONT color="green">161</FONT>         * @param f the complex data array to be transformed<a name="line.161"></a>
<FONT color="green">162</FONT>         * @return the complex transformed array<a name="line.162"></a>
<FONT color="green">163</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.163"></a>
<FONT color="green">164</FONT>         */<a name="line.164"></a>
<FONT color="green">165</FONT>        public Complex[] transform2(Complex f[])<a name="line.165"></a>
<FONT color="green">166</FONT>            throws IllegalArgumentException {<a name="line.166"></a>
<FONT color="green">167</FONT>    <a name="line.167"></a>
<FONT color="green">168</FONT>            roots.computeOmega(f.length);<a name="line.168"></a>
<FONT color="green">169</FONT>            double scaling_coefficient = 1.0 / Math.sqrt(f.length);<a name="line.169"></a>
<FONT color="green">170</FONT>            return scaleArray(fft(f), scaling_coefficient);<a name="line.170"></a>
<FONT color="green">171</FONT>        }<a name="line.171"></a>
<FONT color="green">172</FONT>    <a name="line.172"></a>
<FONT color="green">173</FONT>        /**<a name="line.173"></a>
<FONT color="green">174</FONT>         * Inversely transform the given real data set.<a name="line.174"></a>
<FONT color="green">175</FONT>         * &lt;p&gt;<a name="line.175"></a>
<FONT color="green">176</FONT>         * The formula is $ x_k = (1/N) \Sigma_{n=0}^{N-1} e^{2 \pi i nk/N} y_n $<a name="line.176"></a>
<FONT color="green">177</FONT>         * &lt;/p&gt;<a name="line.177"></a>
<FONT color="green">178</FONT>         * <a name="line.178"></a>
<FONT color="green">179</FONT>         * @param f the real data array to be inversely transformed<a name="line.179"></a>
<FONT color="green">180</FONT>         * @return the complex inversely transformed array<a name="line.180"></a>
<FONT color="green">181</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.181"></a>
<FONT color="green">182</FONT>         */<a name="line.182"></a>
<FONT color="green">183</FONT>        public Complex[] inversetransform(double f[])<a name="line.183"></a>
<FONT color="green">184</FONT>            throws IllegalArgumentException {<a name="line.184"></a>
<FONT color="green">185</FONT>    <a name="line.185"></a>
<FONT color="green">186</FONT>            double scaling_coefficient = 1.0 / f.length;<a name="line.186"></a>
<FONT color="green">187</FONT>            return scaleArray(fft(f, true), scaling_coefficient);<a name="line.187"></a>
<FONT color="green">188</FONT>        }<a name="line.188"></a>
<FONT color="green">189</FONT>    <a name="line.189"></a>
<FONT color="green">190</FONT>        /**<a name="line.190"></a>
<FONT color="green">191</FONT>         * Inversely transform the given real function, sampled on the given interval.<a name="line.191"></a>
<FONT color="green">192</FONT>         * &lt;p&gt;<a name="line.192"></a>
<FONT color="green">193</FONT>         * The formula is $ x_k = (1/N) \Sigma_{n=0}^{N-1} e^{2 \pi i nk/N} y_n $<a name="line.193"></a>
<FONT color="green">194</FONT>         * &lt;/p&gt;<a name="line.194"></a>
<FONT color="green">195</FONT>         * <a name="line.195"></a>
<FONT color="green">196</FONT>         * @param f the function to be sampled and inversely transformed<a name="line.196"></a>
<FONT color="green">197</FONT>         * @param min the lower bound for the interval<a name="line.197"></a>
<FONT color="green">198</FONT>         * @param max the upper bound for the interval<a name="line.198"></a>
<FONT color="green">199</FONT>         * @param n the number of sample points<a name="line.199"></a>
<FONT color="green">200</FONT>         * @return the complex inversely transformed array<a name="line.200"></a>
<FONT color="green">201</FONT>         * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.201"></a>
<FONT color="green">202</FONT>         * at some point<a name="line.202"></a>
<FONT color="green">203</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.203"></a>
<FONT color="green">204</FONT>         */<a name="line.204"></a>
<FONT color="green">205</FONT>        public Complex[] inversetransform(UnivariateRealFunction f,<a name="line.205"></a>
<FONT color="green">206</FONT>                                          double min, double max, int n)<a name="line.206"></a>
<FONT color="green">207</FONT>            throws FunctionEvaluationException, IllegalArgumentException {<a name="line.207"></a>
<FONT color="green">208</FONT>    <a name="line.208"></a>
<FONT color="green">209</FONT>            double data[] = sample(f, min, max, n);<a name="line.209"></a>
<FONT color="green">210</FONT>            double scaling_coefficient = 1.0 / n;<a name="line.210"></a>
<FONT color="green">211</FONT>            return scaleArray(fft(data, true), scaling_coefficient);<a name="line.211"></a>
<FONT color="green">212</FONT>        }<a name="line.212"></a>
<FONT color="green">213</FONT>    <a name="line.213"></a>
<FONT color="green">214</FONT>        /**<a name="line.214"></a>
<FONT color="green">215</FONT>         * Inversely transform the given complex data set.<a name="line.215"></a>
<FONT color="green">216</FONT>         * &lt;p&gt;<a name="line.216"></a>
<FONT color="green">217</FONT>         * The formula is $ x_k = (1/N) \Sigma_{n=0}^{N-1} e^{2 \pi i nk/N} y_n $<a name="line.217"></a>
<FONT color="green">218</FONT>         * &lt;/p&gt;<a name="line.218"></a>
<FONT color="green">219</FONT>         * <a name="line.219"></a>
<FONT color="green">220</FONT>         * @param f the complex data array to be inversely transformed<a name="line.220"></a>
<FONT color="green">221</FONT>         * @return the complex inversely transformed array<a name="line.221"></a>
<FONT color="green">222</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.222"></a>
<FONT color="green">223</FONT>         */<a name="line.223"></a>
<FONT color="green">224</FONT>        public Complex[] inversetransform(Complex f[])<a name="line.224"></a>
<FONT color="green">225</FONT>            throws IllegalArgumentException {<a name="line.225"></a>
<FONT color="green">226</FONT>    <a name="line.226"></a>
<FONT color="green">227</FONT>            roots.computeOmega(-f.length);    // pass negative argument<a name="line.227"></a>
<FONT color="green">228</FONT>            double scaling_coefficient = 1.0 / f.length;<a name="line.228"></a>
<FONT color="green">229</FONT>            return scaleArray(fft(f), scaling_coefficient);<a name="line.229"></a>
<FONT color="green">230</FONT>        }<a name="line.230"></a>
<FONT color="green">231</FONT>    <a name="line.231"></a>
<FONT color="green">232</FONT>        /**<a name="line.232"></a>
<FONT color="green">233</FONT>         * Inversely transform the given real data set.<a name="line.233"></a>
<FONT color="green">234</FONT>         * &lt;p&gt;<a name="line.234"></a>
<FONT color="green">235</FONT>         * The formula is $x_k = (1/\sqrt{N}) \Sigma_{n=0}^{N-1} e^{2 \pi i nk/N} y_n$<a name="line.235"></a>
<FONT color="green">236</FONT>         * &lt;/p&gt;<a name="line.236"></a>
<FONT color="green">237</FONT>         * <a name="line.237"></a>
<FONT color="green">238</FONT>         * @param f the real data array to be inversely transformed<a name="line.238"></a>
<FONT color="green">239</FONT>         * @return the complex inversely transformed array<a name="line.239"></a>
<FONT color="green">240</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.240"></a>
<FONT color="green">241</FONT>         */<a name="line.241"></a>
<FONT color="green">242</FONT>        public Complex[] inversetransform2(double f[])<a name="line.242"></a>
<FONT color="green">243</FONT>            throws IllegalArgumentException {<a name="line.243"></a>
<FONT color="green">244</FONT>    <a name="line.244"></a>
<FONT color="green">245</FONT>            double scaling_coefficient = 1.0 / Math.sqrt(f.length);<a name="line.245"></a>
<FONT color="green">246</FONT>            return scaleArray(fft(f, true), scaling_coefficient);<a name="line.246"></a>
<FONT color="green">247</FONT>        }<a name="line.247"></a>
<FONT color="green">248</FONT>    <a name="line.248"></a>
<FONT color="green">249</FONT>        /**<a name="line.249"></a>
<FONT color="green">250</FONT>         * Inversely transform the given real function, sampled on the given interval.<a name="line.250"></a>
<FONT color="green">251</FONT>         * &lt;p&gt;<a name="line.251"></a>
<FONT color="green">252</FONT>         * The formula is $x_k = (1/\sqrt{N}) \Sigma_{n=0}^{N-1} e^{2 \pi i nk/N} y_n$<a name="line.252"></a>
<FONT color="green">253</FONT>         * &lt;/p&gt;<a name="line.253"></a>
<FONT color="green">254</FONT>         * <a name="line.254"></a>
<FONT color="green">255</FONT>         * @param f the function to be sampled and inversely transformed<a name="line.255"></a>
<FONT color="green">256</FONT>         * @param min the lower bound for the interval<a name="line.256"></a>
<FONT color="green">257</FONT>         * @param max the upper bound for the interval<a name="line.257"></a>
<FONT color="green">258</FONT>         * @param n the number of sample points<a name="line.258"></a>
<FONT color="green">259</FONT>         * @return the complex inversely transformed array<a name="line.259"></a>
<FONT color="green">260</FONT>         * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.260"></a>
<FONT color="green">261</FONT>         * at some point<a name="line.261"></a>
<FONT color="green">262</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.262"></a>
<FONT color="green">263</FONT>         */<a name="line.263"></a>
<FONT color="green">264</FONT>        public Complex[] inversetransform2(UnivariateRealFunction f,<a name="line.264"></a>
<FONT color="green">265</FONT>                                           double min, double max, int n)<a name="line.265"></a>
<FONT color="green">266</FONT>            throws FunctionEvaluationException, IllegalArgumentException {<a name="line.266"></a>
<FONT color="green">267</FONT>    <a name="line.267"></a>
<FONT color="green">268</FONT>            double data[] = sample(f, min, max, n);<a name="line.268"></a>
<FONT color="green">269</FONT>            double scaling_coefficient = 1.0 / Math.sqrt(n);<a name="line.269"></a>
<FONT color="green">270</FONT>            return scaleArray(fft(data, true), scaling_coefficient);<a name="line.270"></a>
<FONT color="green">271</FONT>        }<a name="line.271"></a>
<FONT color="green">272</FONT>    <a name="line.272"></a>
<FONT color="green">273</FONT>        /**<a name="line.273"></a>
<FONT color="green">274</FONT>         * Inversely transform the given complex data set.<a name="line.274"></a>
<FONT color="green">275</FONT>         * &lt;p&gt;<a name="line.275"></a>
<FONT color="green">276</FONT>         * The formula is $x_k = (1/\sqrt{N}) \Sigma_{n=0}^{N-1} e^{2 \pi i nk/N} y_n$<a name="line.276"></a>
<FONT color="green">277</FONT>         * &lt;/p&gt;<a name="line.277"></a>
<FONT color="green">278</FONT>         * <a name="line.278"></a>
<FONT color="green">279</FONT>         * @param f the complex data array to be inversely transformed<a name="line.279"></a>
<FONT color="green">280</FONT>         * @return the complex inversely transformed array<a name="line.280"></a>
<FONT color="green">281</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.281"></a>
<FONT color="green">282</FONT>         */<a name="line.282"></a>
<FONT color="green">283</FONT>        public Complex[] inversetransform2(Complex f[])<a name="line.283"></a>
<FONT color="green">284</FONT>            throws IllegalArgumentException {<a name="line.284"></a>
<FONT color="green">285</FONT>    <a name="line.285"></a>
<FONT color="green">286</FONT>            roots.computeOmega(-f.length);    // pass negative argument<a name="line.286"></a>
<FONT color="green">287</FONT>            double scaling_coefficient = 1.0 / Math.sqrt(f.length);<a name="line.287"></a>
<FONT color="green">288</FONT>            return scaleArray(fft(f), scaling_coefficient);<a name="line.288"></a>
<FONT color="green">289</FONT>        }<a name="line.289"></a>
<FONT color="green">290</FONT>    <a name="line.290"></a>
<FONT color="green">291</FONT>        /**<a name="line.291"></a>
<FONT color="green">292</FONT>         * Perform the base-4 Cooley-Tukey FFT algorithm (including inverse).<a name="line.292"></a>
<FONT color="green">293</FONT>         *<a name="line.293"></a>
<FONT color="green">294</FONT>         * @param f the real data array to be transformed<a name="line.294"></a>
<FONT color="green">295</FONT>         * @param isInverse the indicator of forward or inverse transform<a name="line.295"></a>
<FONT color="green">296</FONT>         * @return the complex transformed array<a name="line.296"></a>
<FONT color="green">297</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.297"></a>
<FONT color="green">298</FONT>         */<a name="line.298"></a>
<FONT color="green">299</FONT>        protected Complex[] fft(double f[], boolean isInverse)<a name="line.299"></a>
<FONT color="green">300</FONT>            throws IllegalArgumentException {<a name="line.300"></a>
<FONT color="green">301</FONT>    <a name="line.301"></a>
<FONT color="green">302</FONT>            verifyDataSet(f);<a name="line.302"></a>
<FONT color="green">303</FONT>            Complex F[] = new Complex[f.length];<a name="line.303"></a>
<FONT color="green">304</FONT>            if (f.length == 1) {<a name="line.304"></a>
<FONT color="green">305</FONT>                F[0] = new Complex(f[0], 0.0);<a name="line.305"></a>
<FONT color="green">306</FONT>                return F;<a name="line.306"></a>
<FONT color="green">307</FONT>            }<a name="line.307"></a>
<FONT color="green">308</FONT>    <a name="line.308"></a>
<FONT color="green">309</FONT>            // Rather than the naive real to complex conversion, pack 2N<a name="line.309"></a>
<FONT color="green">310</FONT>            // real numbers into N complex numbers for better performance.<a name="line.310"></a>
<FONT color="green">311</FONT>            int N = f.length &gt;&gt; 1;<a name="line.311"></a>
<FONT color="green">312</FONT>            Complex c[] = new Complex[N];<a name="line.312"></a>
<FONT color="green">313</FONT>            for (int i = 0; i &lt; N; i++) {<a name="line.313"></a>
<FONT color="green">314</FONT>                c[i] = new Complex(f[2*i], f[2*i+1]);<a name="line.314"></a>
<FONT color="green">315</FONT>            }<a name="line.315"></a>
<FONT color="green">316</FONT>            roots.computeOmega(isInverse ? -N : N);<a name="line.316"></a>
<FONT color="green">317</FONT>            Complex z[] = fft(c);<a name="line.317"></a>
<FONT color="green">318</FONT>    <a name="line.318"></a>
<FONT color="green">319</FONT>            // reconstruct the FFT result for the original array<a name="line.319"></a>
<FONT color="green">320</FONT>            roots.computeOmega(isInverse ? -2*N : 2*N);<a name="line.320"></a>
<FONT color="green">321</FONT>            F[0] = new Complex(2 * (z[0].getReal() + z[0].getImaginary()), 0.0);<a name="line.321"></a>
<FONT color="green">322</FONT>            F[N] = new Complex(2 * (z[0].getReal() - z[0].getImaginary()), 0.0);<a name="line.322"></a>
<FONT color="green">323</FONT>            for (int i = 1; i &lt; N; i++) {<a name="line.323"></a>
<FONT color="green">324</FONT>                Complex A = z[N-i].conjugate();<a name="line.324"></a>
<FONT color="green">325</FONT>                Complex B = z[i].add(A);<a name="line.325"></a>
<FONT color="green">326</FONT>                Complex C = z[i].subtract(A);<a name="line.326"></a>
<FONT color="green">327</FONT>                //Complex D = roots.getOmega(i).multiply(Complex.I);<a name="line.327"></a>
<FONT color="green">328</FONT>                Complex D = new Complex(-roots.getOmegaImaginary(i),<a name="line.328"></a>
<FONT color="green">329</FONT>                                        roots.getOmegaReal(i));<a name="line.329"></a>
<FONT color="green">330</FONT>                F[i] = B.subtract(C.multiply(D));<a name="line.330"></a>
<FONT color="green">331</FONT>                F[2*N-i] = F[i].conjugate();<a name="line.331"></a>
<FONT color="green">332</FONT>            }<a name="line.332"></a>
<FONT color="green">333</FONT>    <a name="line.333"></a>
<FONT color="green">334</FONT>            return scaleArray(F, 0.5);<a name="line.334"></a>
<FONT color="green">335</FONT>        }<a name="line.335"></a>
<FONT color="green">336</FONT>    <a name="line.336"></a>
<FONT color="green">337</FONT>        /**<a name="line.337"></a>
<FONT color="green">338</FONT>         * Perform the base-4 Cooley-Tukey FFT algorithm (including inverse).<a name="line.338"></a>
<FONT color="green">339</FONT>         *<a name="line.339"></a>
<FONT color="green">340</FONT>         * @param data the complex data array to be transformed<a name="line.340"></a>
<FONT color="green">341</FONT>         * @return the complex transformed array<a name="line.341"></a>
<FONT color="green">342</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.342"></a>
<FONT color="green">343</FONT>         */<a name="line.343"></a>
<FONT color="green">344</FONT>        protected Complex[] fft(Complex data[])<a name="line.344"></a>
<FONT color="green">345</FONT>            throws IllegalArgumentException {<a name="line.345"></a>
<FONT color="green">346</FONT>    <a name="line.346"></a>
<FONT color="green">347</FONT>            int i, j, k, m, N = data.length;<a name="line.347"></a>
<FONT color="green">348</FONT>            Complex A, B, C, D, E, F, z, f[] = new Complex[N];<a name="line.348"></a>
<FONT color="green">349</FONT>    <a name="line.349"></a>
<FONT color="green">350</FONT>            // initial simple cases<a name="line.350"></a>
<FONT color="green">351</FONT>            verifyDataSet(data);<a name="line.351"></a>
<FONT color="green">352</FONT>            if (N == 1) {<a name="line.352"></a>
<FONT color="green">353</FONT>                f[0] = data[0];<a name="line.353"></a>
<FONT color="green">354</FONT>                return f;<a name="line.354"></a>
<FONT color="green">355</FONT>            }<a name="line.355"></a>
<FONT color="green">356</FONT>            if (N == 2) {<a name="line.356"></a>
<FONT color="green">357</FONT>                f[0] = data[0].add(data[1]);<a name="line.357"></a>
<FONT color="green">358</FONT>                f[1] = data[0].subtract(data[1]);<a name="line.358"></a>
<FONT color="green">359</FONT>                return f;<a name="line.359"></a>
<FONT color="green">360</FONT>            }<a name="line.360"></a>
<FONT color="green">361</FONT>    <a name="line.361"></a>
<FONT color="green">362</FONT>            // permute original data array in bit-reversal order<a name="line.362"></a>
<FONT color="green">363</FONT>            j = 0;<a name="line.363"></a>
<FONT color="green">364</FONT>            for (i = 0; i &lt; N; i++) {<a name="line.364"></a>
<FONT color="green">365</FONT>                f[i] = data[j];<a name="line.365"></a>
<FONT color="green">366</FONT>                k = N &gt;&gt; 1;<a name="line.366"></a>
<FONT color="green">367</FONT>                while (j &gt;= k &amp;&amp; k &gt; 0) {<a name="line.367"></a>
<FONT color="green">368</FONT>                    j -= k; k &gt;&gt;= 1;<a name="line.368"></a>
<FONT color="green">369</FONT>                }<a name="line.369"></a>
<FONT color="green">370</FONT>                j += k;<a name="line.370"></a>
<FONT color="green">371</FONT>            }<a name="line.371"></a>
<FONT color="green">372</FONT>    <a name="line.372"></a>
<FONT color="green">373</FONT>            // the bottom base-4 round<a name="line.373"></a>
<FONT color="green">374</FONT>            for (i = 0; i &lt; N; i += 4) {<a name="line.374"></a>
<FONT color="green">375</FONT>                A = f[i].add(f[i+1]);<a name="line.375"></a>
<FONT color="green">376</FONT>                B = f[i+2].add(f[i+3]);<a name="line.376"></a>
<FONT color="green">377</FONT>                C = f[i].subtract(f[i+1]);<a name="line.377"></a>
<FONT color="green">378</FONT>                D = f[i+2].subtract(f[i+3]);<a name="line.378"></a>
<FONT color="green">379</FONT>                E = C.add(D.multiply(Complex.I));<a name="line.379"></a>
<FONT color="green">380</FONT>                F = C.subtract(D.multiply(Complex.I));<a name="line.380"></a>
<FONT color="green">381</FONT>                f[i] = A.add(B);<a name="line.381"></a>
<FONT color="green">382</FONT>                f[i+2] = A.subtract(B);<a name="line.382"></a>
<FONT color="green">383</FONT>                // omegaCount indicates forward or inverse transform<a name="line.383"></a>
<FONT color="green">384</FONT>                f[i+1] = roots.isForward() ? F : E;<a name="line.384"></a>
<FONT color="green">385</FONT>                f[i+3] = roots.isForward() ? E : F;<a name="line.385"></a>
<FONT color="green">386</FONT>            }<a name="line.386"></a>
<FONT color="green">387</FONT>    <a name="line.387"></a>
<FONT color="green">388</FONT>            // iterations from bottom to top take O(N*logN) time<a name="line.388"></a>
<FONT color="green">389</FONT>            for (i = 4; i &lt; N; i &lt;&lt;= 1) {<a name="line.389"></a>
<FONT color="green">390</FONT>                m = N / (i&lt;&lt;1);<a name="line.390"></a>
<FONT color="green">391</FONT>                for (j = 0; j &lt; N; j += i&lt;&lt;1) {<a name="line.391"></a>
<FONT color="green">392</FONT>                    for (k = 0; k &lt; i; k++) {<a name="line.392"></a>
<FONT color="green">393</FONT>                        //z = f[i+j+k].multiply(roots.getOmega(k*m));<a name="line.393"></a>
<FONT color="green">394</FONT>                        final int k_times_m = k*m;<a name="line.394"></a>
<FONT color="green">395</FONT>                        final double omega_k_times_m_real = roots.getOmegaReal(k_times_m);<a name="line.395"></a>
<FONT color="green">396</FONT>                        final double omega_k_times_m_imaginary = roots.getOmegaImaginary(k_times_m);<a name="line.396"></a>
<FONT color="green">397</FONT>                        //z = f[i+j+k].multiply(omega[k*m]);<a name="line.397"></a>
<FONT color="green">398</FONT>                        z = new Complex(<a name="line.398"></a>
<FONT color="green">399</FONT>                            f[i+j+k].getReal() * omega_k_times_m_real -<a name="line.399"></a>
<FONT color="green">400</FONT>                            f[i+j+k].getImaginary() * omega_k_times_m_imaginary,<a name="line.400"></a>
<FONT color="green">401</FONT>                            f[i+j+k].getReal() * omega_k_times_m_imaginary +<a name="line.401"></a>
<FONT color="green">402</FONT>                            f[i+j+k].getImaginary() * omega_k_times_m_real);<a name="line.402"></a>
<FONT color="green">403</FONT>                      <a name="line.403"></a>
<FONT color="green">404</FONT>                        f[i+j+k] = f[j+k].subtract(z);<a name="line.404"></a>
<FONT color="green">405</FONT>                        f[j+k] = f[j+k].add(z);<a name="line.405"></a>
<FONT color="green">406</FONT>                    }<a name="line.406"></a>
<FONT color="green">407</FONT>                }<a name="line.407"></a>
<FONT color="green">408</FONT>            }<a name="line.408"></a>
<FONT color="green">409</FONT>            return f;<a name="line.409"></a>
<FONT color="green">410</FONT>        }<a name="line.410"></a>
<FONT color="green">411</FONT>    <a name="line.411"></a>
<FONT color="green">412</FONT>        /**<a name="line.412"></a>
<FONT color="green">413</FONT>         * Sample the given univariate real function on the given interval.<a name="line.413"></a>
<FONT color="green">414</FONT>         * &lt;p&gt;<a name="line.414"></a>
<FONT color="green">415</FONT>         * The interval is divided equally into N sections and sample points<a name="line.415"></a>
<FONT color="green">416</FONT>         * are taken from min to max-(max-min)/N. Usually f(x) is periodic<a name="line.416"></a>
<FONT color="green">417</FONT>         * such that f(min) = f(max) (note max is not sampled), but we don't<a name="line.417"></a>
<FONT color="green">418</FONT>         * require that.&lt;/p&gt;<a name="line.418"></a>
<FONT color="green">419</FONT>         *<a name="line.419"></a>
<FONT color="green">420</FONT>         * @param f the function to be sampled<a name="line.420"></a>
<FONT color="green">421</FONT>         * @param min the lower bound for the interval<a name="line.421"></a>
<FONT color="green">422</FONT>         * @param max the upper bound for the interval<a name="line.422"></a>
<FONT color="green">423</FONT>         * @param n the number of sample points<a name="line.423"></a>
<FONT color="green">424</FONT>         * @return the samples array<a name="line.424"></a>
<FONT color="green">425</FONT>         * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.425"></a>
<FONT color="green">426</FONT>         * at some point<a name="line.426"></a>
<FONT color="green">427</FONT>         * @throws IllegalArgumentException if any parameters are invalid<a name="line.427"></a>
<FONT color="green">428</FONT>         */<a name="line.428"></a>
<FONT color="green">429</FONT>        public static double[] sample(UnivariateRealFunction f,<a name="line.429"></a>
<FONT color="green">430</FONT>                                      double min, double max, int n)<a name="line.430"></a>
<FONT color="green">431</FONT>            throws FunctionEvaluationException, IllegalArgumentException {<a name="line.431"></a>
<FONT color="green">432</FONT>    <a name="line.432"></a>
<FONT color="green">433</FONT>            if (n &lt;= 0) {<a name="line.433"></a>
<FONT color="green">434</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.434"></a>
<FONT color="green">435</FONT>                        "number of sample is not positive: {0}",<a name="line.435"></a>
<FONT color="green">436</FONT>                        n);<a name="line.436"></a>
<FONT color="green">437</FONT>            }<a name="line.437"></a>
<FONT color="green">438</FONT>            verifyInterval(min, max);<a name="line.438"></a>
<FONT color="green">439</FONT>    <a name="line.439"></a>
<FONT color="green">440</FONT>            double s[] = new double[n];<a name="line.440"></a>
<FONT color="green">441</FONT>            double h = (max - min) / n;<a name="line.441"></a>
<FONT color="green">442</FONT>            for (int i = 0; i &lt; n; i++) {<a name="line.442"></a>
<FONT color="green">443</FONT>                s[i] = f.value(min + i * h);<a name="line.443"></a>
<FONT color="green">444</FONT>            }<a name="line.444"></a>
<FONT color="green">445</FONT>            return s;<a name="line.445"></a>
<FONT color="green">446</FONT>        }<a name="line.446"></a>
<FONT color="green">447</FONT>    <a name="line.447"></a>
<FONT color="green">448</FONT>        /**<a name="line.448"></a>
<FONT color="green">449</FONT>         * Multiply every component in the given real array by the<a name="line.449"></a>
<FONT color="green">450</FONT>         * given real number. The change is made in place.<a name="line.450"></a>
<FONT color="green">451</FONT>         *<a name="line.451"></a>
<FONT color="green">452</FONT>         * @param f the real array to be scaled<a name="line.452"></a>
<FONT color="green">453</FONT>         * @param d the real scaling coefficient<a name="line.453"></a>
<FONT color="green">454</FONT>         * @return a reference to the scaled array<a name="line.454"></a>
<FONT color="green">455</FONT>         */<a name="line.455"></a>
<FONT color="green">456</FONT>        public static double[] scaleArray(double f[], double d) {<a name="line.456"></a>
<FONT color="green">457</FONT>            for (int i = 0; i &lt; f.length; i++) {<a name="line.457"></a>
<FONT color="green">458</FONT>                f[i] *= d;<a name="line.458"></a>
<FONT color="green">459</FONT>            }<a name="line.459"></a>
<FONT color="green">460</FONT>            return f;<a name="line.460"></a>
<FONT color="green">461</FONT>        }<a name="line.461"></a>
<FONT color="green">462</FONT>    <a name="line.462"></a>
<FONT color="green">463</FONT>        /**<a name="line.463"></a>
<FONT color="green">464</FONT>         * Multiply every component in the given complex array by the<a name="line.464"></a>
<FONT color="green">465</FONT>         * given real number. The change is made in place.<a name="line.465"></a>
<FONT color="green">466</FONT>         *<a name="line.466"></a>
<FONT color="green">467</FONT>         * @param f the complex array to be scaled<a name="line.467"></a>
<FONT color="green">468</FONT>         * @param d the real scaling coefficient<a name="line.468"></a>
<FONT color="green">469</FONT>         * @return a reference to the scaled array<a name="line.469"></a>
<FONT color="green">470</FONT>         */<a name="line.470"></a>
<FONT color="green">471</FONT>        public static Complex[] scaleArray(Complex f[], double d) {<a name="line.471"></a>
<FONT color="green">472</FONT>            for (int i = 0; i &lt; f.length; i++) {<a name="line.472"></a>
<FONT color="green">473</FONT>                f[i] = new Complex(d * f[i].getReal(), d * f[i].getImaginary());<a name="line.473"></a>
<FONT color="green">474</FONT>            }<a name="line.474"></a>
<FONT color="green">475</FONT>            return f;<a name="line.475"></a>
<FONT color="green">476</FONT>        }<a name="line.476"></a>
<FONT color="green">477</FONT>    <a name="line.477"></a>
<FONT color="green">478</FONT>        /**<a name="line.478"></a>
<FONT color="green">479</FONT>         * Returns true if the argument is power of 2.<a name="line.479"></a>
<FONT color="green">480</FONT>         * <a name="line.480"></a>
<FONT color="green">481</FONT>         * @param n the number to test<a name="line.481"></a>
<FONT color="green">482</FONT>         * @return true if the argument is power of 2<a name="line.482"></a>
<FONT color="green">483</FONT>         */<a name="line.483"></a>
<FONT color="green">484</FONT>        public static boolean isPowerOf2(long n) {<a name="line.484"></a>
<FONT color="green">485</FONT>            return (n &gt; 0) &amp;&amp; ((n &amp; (n - 1)) == 0);<a name="line.485"></a>
<FONT color="green">486</FONT>        }<a name="line.486"></a>
<FONT color="green">487</FONT>    <a name="line.487"></a>
<FONT color="green">488</FONT>        /**<a name="line.488"></a>
<FONT color="green">489</FONT>         * Verifies that the data set has length of power of 2.<a name="line.489"></a>
<FONT color="green">490</FONT>         * <a name="line.490"></a>
<FONT color="green">491</FONT>         * @param d the data array<a name="line.491"></a>
<FONT color="green">492</FONT>         * @throws IllegalArgumentException if array length is not power of 2<a name="line.492"></a>
<FONT color="green">493</FONT>         */<a name="line.493"></a>
<FONT color="green">494</FONT>        public static void verifyDataSet(double d[]) throws IllegalArgumentException {<a name="line.494"></a>
<FONT color="green">495</FONT>            if (!isPowerOf2(d.length)) {<a name="line.495"></a>
<FONT color="green">496</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.496"></a>
<FONT color="green">497</FONT>                        "{0} is not a power of 2, consider padding for fix",<a name="line.497"></a>
<FONT color="green">498</FONT>                        d.length);<a name="line.498"></a>
<FONT color="green">499</FONT>            }       <a name="line.499"></a>
<FONT color="green">500</FONT>        }<a name="line.500"></a>
<FONT color="green">501</FONT>    <a name="line.501"></a>
<FONT color="green">502</FONT>        /**<a name="line.502"></a>
<FONT color="green">503</FONT>         * Verifies that the data set has length of power of 2.<a name="line.503"></a>
<FONT color="green">504</FONT>         * <a name="line.504"></a>
<FONT color="green">505</FONT>         * @param o the data array<a name="line.505"></a>
<FONT color="green">506</FONT>         * @throws IllegalArgumentException if array length is not power of 2<a name="line.506"></a>
<FONT color="green">507</FONT>         */<a name="line.507"></a>
<FONT color="green">508</FONT>        public static void verifyDataSet(Object o[]) throws IllegalArgumentException {<a name="line.508"></a>
<FONT color="green">509</FONT>            if (!isPowerOf2(o.length)) {<a name="line.509"></a>
<FONT color="green">510</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.510"></a>
<FONT color="green">511</FONT>                        "{0} is not a power of 2, consider padding for fix",<a name="line.511"></a>
<FONT color="green">512</FONT>                        o.length);<a name="line.512"></a>
<FONT color="green">513</FONT>            }       <a name="line.513"></a>
<FONT color="green">514</FONT>        }<a name="line.514"></a>
<FONT color="green">515</FONT>    <a name="line.515"></a>
<FONT color="green">516</FONT>        /**<a name="line.516"></a>
<FONT color="green">517</FONT>         * Verifies that the endpoints specify an interval.<a name="line.517"></a>
<FONT color="green">518</FONT>         * <a name="line.518"></a>
<FONT color="green">519</FONT>         * @param lower lower endpoint<a name="line.519"></a>
<FONT color="green">520</FONT>         * @param upper upper endpoint<a name="line.520"></a>
<FONT color="green">521</FONT>         * @throws IllegalArgumentException if not interval<a name="line.521"></a>
<FONT color="green">522</FONT>         */<a name="line.522"></a>
<FONT color="green">523</FONT>        public static void verifyInterval(double lower, double upper)<a name="line.523"></a>
<FONT color="green">524</FONT>            throws IllegalArgumentException {<a name="line.524"></a>
<FONT color="green">525</FONT>    <a name="line.525"></a>
<FONT color="green">526</FONT>            if (lower &gt;= upper) {<a name="line.526"></a>
<FONT color="green">527</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.527"></a>
<FONT color="green">528</FONT>                        "endpoints do not specify an interval: [{0}, {1}]",<a name="line.528"></a>
<FONT color="green">529</FONT>                        lower, upper);<a name="line.529"></a>
<FONT color="green">530</FONT>            }       <a name="line.530"></a>
<FONT color="green">531</FONT>        }<a name="line.531"></a>
<FONT color="green">532</FONT>        <a name="line.532"></a>
<FONT color="green">533</FONT>        /**<a name="line.533"></a>
<FONT color="green">534</FONT>         * Performs a multi-dimensional Fourier transform on a given array.<a name="line.534"></a>
<FONT color="green">535</FONT>         * Use {@link #inversetransform2(Complex[])} and<a name="line.535"></a>
<FONT color="green">536</FONT>         * {@link #transform2(Complex[])} in a row-column implementation<a name="line.536"></a>
<FONT color="green">537</FONT>         * in any number of dimensions with O(N&amp;times;log(N)) complexity with<a name="line.537"></a>
<FONT color="green">538</FONT>         * N=n&lt;sub&gt;1&lt;/sub&gt;&amp;times;n&lt;sub&gt;2&lt;/sub&gt;&amp;times;n&lt;sub&gt;3&lt;/sub&gt;&amp;times;...&amp;times;n&lt;sub&gt;d&lt;/sub&gt;,<a name="line.538"></a>
<FONT color="green">539</FONT>         * n&lt;sub&gt;x&lt;/sub&gt;=number of elements in dimension x,<a name="line.539"></a>
<FONT color="green">540</FONT>         * and d=total number of dimensions.<a name="line.540"></a>
<FONT color="green">541</FONT>         *<a name="line.541"></a>
<FONT color="green">542</FONT>         * @param mdca Multi-Dimensional Complex Array id est Complex[][][][]<a name="line.542"></a>
<FONT color="green">543</FONT>         * @param forward inverseTransform2 is preformed if this is false<a name="line.543"></a>
<FONT color="green">544</FONT>         * @return transform of mdca as a Multi-Dimensional Complex Array id est Complex[][][][]<a name="line.544"></a>
<FONT color="green">545</FONT>         * @throws IllegalArgumentException if any dimension is not a power of two<a name="line.545"></a>
<FONT color="green">546</FONT>         */<a name="line.546"></a>
<FONT color="green">547</FONT>        public Object mdfft(Object mdca, boolean forward)<a name="line.547"></a>
<FONT color="green">548</FONT>            throws IllegalArgumentException {<a name="line.548"></a>
<FONT color="green">549</FONT>            MultiDimensionalComplexMatrix mdcm = (MultiDimensionalComplexMatrix)<a name="line.549"></a>
<FONT color="green">550</FONT>                    new MultiDimensionalComplexMatrix(mdca).clone();<a name="line.550"></a>
<FONT color="green">551</FONT>            int[] dimensionSize = mdcm.getDimensionSizes();<a name="line.551"></a>
<FONT color="green">552</FONT>            //cycle through each dimension<a name="line.552"></a>
<FONT color="green">553</FONT>            for (int i = 0; i &lt; dimensionSize.length; i++) {<a name="line.553"></a>
<FONT color="green">554</FONT>                mdfft(mdcm, forward, i, new int[0]);<a name="line.554"></a>
<FONT color="green">555</FONT>            }<a name="line.555"></a>
<FONT color="green">556</FONT>            return mdcm.getArray();<a name="line.556"></a>
<FONT color="green">557</FONT>        }<a name="line.557"></a>
<FONT color="green">558</FONT>        <a name="line.558"></a>
<FONT color="green">559</FONT>        /**<a name="line.559"></a>
<FONT color="green">560</FONT>         * Performs one dimension of a multi-dimensional Fourier transform.<a name="line.560"></a>
<FONT color="green">561</FONT>         *<a name="line.561"></a>
<FONT color="green">562</FONT>         * @param mdcm input matrix<a name="line.562"></a>
<FONT color="green">563</FONT>         * @param forward inverseTransform2 is preformed if this is false<a name="line.563"></a>
<FONT color="green">564</FONT>         * @param d index of the dimension to process<a name="line.564"></a>
<FONT color="green">565</FONT>         * @param subVector recursion subvector<a name="line.565"></a>
<FONT color="green">566</FONT>         * @throws IllegalArgumentException if any dimension is not a power of two<a name="line.566"></a>
<FONT color="green">567</FONT>         */<a name="line.567"></a>
<FONT color="green">568</FONT>        private void mdfft(MultiDimensionalComplexMatrix mdcm, boolean forward,<a name="line.568"></a>
<FONT color="green">569</FONT>                           int d, int[] subVector)<a name="line.569"></a>
<FONT color="green">570</FONT>            throws IllegalArgumentException {<a name="line.570"></a>
<FONT color="green">571</FONT>            int[] dimensionSize = mdcm.getDimensionSizes();<a name="line.571"></a>
<FONT color="green">572</FONT>            //if done<a name="line.572"></a>
<FONT color="green">573</FONT>            if (subVector.length == dimensionSize.length) {<a name="line.573"></a>
<FONT color="green">574</FONT>                Complex[] temp = new Complex[dimensionSize[d]];<a name="line.574"></a>
<FONT color="green">575</FONT>                for (int i = 0; i &lt; dimensionSize[d]; i++) {<a name="line.575"></a>
<FONT color="green">576</FONT>                    //fft along dimension d<a name="line.576"></a>
<FONT color="green">577</FONT>                    subVector[d] = i;<a name="line.577"></a>
<FONT color="green">578</FONT>                    temp[i] = mdcm.get(subVector);<a name="line.578"></a>
<FONT color="green">579</FONT>                }<a name="line.579"></a>
<FONT color="green">580</FONT>                <a name="line.580"></a>
<FONT color="green">581</FONT>                if (forward)<a name="line.581"></a>
<FONT color="green">582</FONT>                    temp = transform2(temp);<a name="line.582"></a>
<FONT color="green">583</FONT>                else<a name="line.583"></a>
<FONT color="green">584</FONT>                    temp = inversetransform2(temp);<a name="line.584"></a>
<FONT color="green">585</FONT>                <a name="line.585"></a>
<FONT color="green">586</FONT>                for (int i = 0; i &lt; dimensionSize[d]; i++) {<a name="line.586"></a>
<FONT color="green">587</FONT>                    subVector[d] = i;<a name="line.587"></a>
<FONT color="green">588</FONT>                    mdcm.set(temp[i], subVector);<a name="line.588"></a>
<FONT color="green">589</FONT>                }<a name="line.589"></a>
<FONT color="green">590</FONT>            } else {<a name="line.590"></a>
<FONT color="green">591</FONT>                int[] vector = new int[subVector.length + 1];<a name="line.591"></a>
<FONT color="green">592</FONT>                System.arraycopy(subVector, 0, vector, 0, subVector.length);<a name="line.592"></a>
<FONT color="green">593</FONT>                if (subVector.length == d) {<a name="line.593"></a>
<FONT color="green">594</FONT>                    //value is not important once the recursion is done.<a name="line.594"></a>
<FONT color="green">595</FONT>                    //then an fft will be applied along the dimension d.<a name="line.595"></a>
<FONT color="green">596</FONT>                    vector[d] = 0;<a name="line.596"></a>
<FONT color="green">597</FONT>                    mdfft(mdcm, forward, d, vector);<a name="line.597"></a>
<FONT color="green">598</FONT>                } else {<a name="line.598"></a>
<FONT color="green">599</FONT>                    for (int i = 0; i &lt; dimensionSize[subVector.length]; i++) {<a name="line.599"></a>
<FONT color="green">600</FONT>                        vector[subVector.length] = i;<a name="line.600"></a>
<FONT color="green">601</FONT>                        //further split along the next dimension<a name="line.601"></a>
<FONT color="green">602</FONT>                        mdfft(mdcm, forward, d, vector);<a name="line.602"></a>
<FONT color="green">603</FONT>                    }<a name="line.603"></a>
<FONT color="green">604</FONT>                }<a name="line.604"></a>
<FONT color="green">605</FONT>            }<a name="line.605"></a>
<FONT color="green">606</FONT>            return;<a name="line.606"></a>
<FONT color="green">607</FONT>        }<a name="line.607"></a>
<FONT color="green">608</FONT>    <a name="line.608"></a>
<FONT color="green">609</FONT>        /**<a name="line.609"></a>
<FONT color="green">610</FONT>         * Complex matrix implementation.<a name="line.610"></a>
<FONT color="green">611</FONT>         * Not designed for synchronized access<a name="line.611"></a>
<FONT color="green">612</FONT>         * may eventually be replaced by jsr-83 of the java community process<a name="line.612"></a>
<FONT color="green">613</FONT>         * http://jcp.org/en/jsr/detail?id=83<a name="line.613"></a>
<FONT color="green">614</FONT>         * may require additional exception throws for other basic requirements.<a name="line.614"></a>
<FONT color="green">615</FONT>         */<a name="line.615"></a>
<FONT color="green">616</FONT>        private static class MultiDimensionalComplexMatrix<a name="line.616"></a>
<FONT color="green">617</FONT>            implements Cloneable {<a name="line.617"></a>
<FONT color="green">618</FONT>    <a name="line.618"></a>
<FONT color="green">619</FONT>            /** Size in all dimensions. */<a name="line.619"></a>
<FONT color="green">620</FONT>            protected int[] dimensionSize;<a name="line.620"></a>
<FONT color="green">621</FONT>    <a name="line.621"></a>
<FONT color="green">622</FONT>            /** Storage array. */<a name="line.622"></a>
<FONT color="green">623</FONT>            protected Object multiDimensionalComplexArray;<a name="line.623"></a>
<FONT color="green">624</FONT>    <a name="line.624"></a>
<FONT color="green">625</FONT>            /** Simple constructor.<a name="line.625"></a>
<FONT color="green">626</FONT>             * @param multiDimensionalComplexArray array containing the matrix elements<a name="line.626"></a>
<FONT color="green">627</FONT>             */<a name="line.627"></a>
<FONT color="green">628</FONT>            public MultiDimensionalComplexMatrix(Object multiDimensionalComplexArray) {<a name="line.628"></a>
<FONT color="green">629</FONT>    <a name="line.629"></a>
<FONT color="green">630</FONT>                this.multiDimensionalComplexArray = multiDimensionalComplexArray;<a name="line.630"></a>
<FONT color="green">631</FONT>    <a name="line.631"></a>
<FONT color="green">632</FONT>                // count dimensions<a name="line.632"></a>
<FONT color="green">633</FONT>                int numOfDimensions = 0;<a name="line.633"></a>
<FONT color="green">634</FONT>                for (Object lastDimension = multiDimensionalComplexArray;<a name="line.634"></a>
<FONT color="green">635</FONT>                     lastDimension instanceof Object[];) {<a name="line.635"></a>
<FONT color="green">636</FONT>                    final Object[] array = (Object[]) lastDimension;<a name="line.636"></a>
<FONT color="green">637</FONT>                    numOfDimensions++;<a name="line.637"></a>
<FONT color="green">638</FONT>                    lastDimension = array[0];<a name="line.638"></a>
<FONT color="green">639</FONT>                }<a name="line.639"></a>
<FONT color="green">640</FONT>    <a name="line.640"></a>
<FONT color="green">641</FONT>                // allocate array with exact count<a name="line.641"></a>
<FONT color="green">642</FONT>                dimensionSize = new int[numOfDimensions];<a name="line.642"></a>
<FONT color="green">643</FONT>    <a name="line.643"></a>
<FONT color="green">644</FONT>                // fill array<a name="line.644"></a>
<FONT color="green">645</FONT>                numOfDimensions = 0;<a name="line.645"></a>
<FONT color="green">646</FONT>                for (Object lastDimension = multiDimensionalComplexArray;<a name="line.646"></a>
<FONT color="green">647</FONT>                     lastDimension instanceof Object[];) {<a name="line.647"></a>
<FONT color="green">648</FONT>                    final Object[] array = (Object[]) lastDimension;<a name="line.648"></a>
<FONT color="green">649</FONT>                    dimensionSize[numOfDimensions++] = array.length;<a name="line.649"></a>
<FONT color="green">650</FONT>                    lastDimension = array[0];<a name="line.650"></a>
<FONT color="green">651</FONT>                }<a name="line.651"></a>
<FONT color="green">652</FONT>    <a name="line.652"></a>
<FONT color="green">653</FONT>            }<a name="line.653"></a>
<FONT color="green">654</FONT>    <a name="line.654"></a>
<FONT color="green">655</FONT>            /**<a name="line.655"></a>
<FONT color="green">656</FONT>             * Get a matrix element.<a name="line.656"></a>
<FONT color="green">657</FONT>             * @param vector indices of the element<a name="line.657"></a>
<FONT color="green">658</FONT>             * @return matrix element<a name="line.658"></a>
<FONT color="green">659</FONT>             * @exception IllegalArgumentException if dimensions do not match<a name="line.659"></a>
<FONT color="green">660</FONT>             */<a name="line.660"></a>
<FONT color="green">661</FONT>            public Complex get(int... vector)<a name="line.661"></a>
<FONT color="green">662</FONT>                throws IllegalArgumentException {<a name="line.662"></a>
<FONT color="green">663</FONT>                if (vector == null) {<a name="line.663"></a>
<FONT color="green">664</FONT>                    if (dimensionSize.length &gt; 0) {<a name="line.664"></a>
<FONT color="green">665</FONT>                        throw MathRuntimeException.createIllegalArgumentException(<a name="line.665"></a>
<FONT color="green">666</FONT>                                "some dimensions don't match: {0} != {1}",<a name="line.666"></a>
<FONT color="green">667</FONT>                                0, dimensionSize.length);<a name="line.667"></a>
<FONT color="green">668</FONT>                    }<a name="line.668"></a>
<FONT color="green">669</FONT>                    return null;<a name="line.669"></a>
<FONT color="green">670</FONT>                }<a name="line.670"></a>
<FONT color="green">671</FONT>                if (vector.length != dimensionSize.length) {<a name="line.671"></a>
<FONT color="green">672</FONT>                    throw MathRuntimeException.createIllegalArgumentException(<a name="line.672"></a>
<FONT color="green">673</FONT>                            "some dimensions don't match: {0} != {1}",<a name="line.673"></a>
<FONT color="green">674</FONT>                            vector.length, dimensionSize.length);<a name="line.674"></a>
<FONT color="green">675</FONT>                }<a name="line.675"></a>
<FONT color="green">676</FONT>                <a name="line.676"></a>
<FONT color="green">677</FONT>                Object lastDimension = multiDimensionalComplexArray;<a name="line.677"></a>
<FONT color="green">678</FONT>                <a name="line.678"></a>
<FONT color="green">679</FONT>                for (int i = 0; i &lt; dimensionSize.length; i++) {<a name="line.679"></a>
<FONT color="green">680</FONT>                    lastDimension = ((Object[]) lastDimension)[vector[i]];<a name="line.680"></a>
<FONT color="green">681</FONT>                }<a name="line.681"></a>
<FONT color="green">682</FONT>                return (Complex) lastDimension;<a name="line.682"></a>
<FONT color="green">683</FONT>            }<a name="line.683"></a>
<FONT color="green">684</FONT>            <a name="line.684"></a>
<FONT color="green">685</FONT>            /**<a name="line.685"></a>
<FONT color="green">686</FONT>             * Set a matrix element.<a name="line.686"></a>
<FONT color="green">687</FONT>             * @param magnitude magnitude of the element<a name="line.687"></a>
<FONT color="green">688</FONT>             * @param vector indices of the element<a name="line.688"></a>
<FONT color="green">689</FONT>             * @return the previous value<a name="line.689"></a>
<FONT color="green">690</FONT>             * @exception IllegalArgumentException if dimensions do not match<a name="line.690"></a>
<FONT color="green">691</FONT>             */<a name="line.691"></a>
<FONT color="green">692</FONT>            public Complex set(Complex magnitude, int... vector)<a name="line.692"></a>
<FONT color="green">693</FONT>                throws IllegalArgumentException {<a name="line.693"></a>
<FONT color="green">694</FONT>                if (vector == null) {<a name="line.694"></a>
<FONT color="green">695</FONT>                    if (dimensionSize.length &gt; 0) {<a name="line.695"></a>
<FONT color="green">696</FONT>                        throw MathRuntimeException.createIllegalArgumentException(<a name="line.696"></a>
<FONT color="green">697</FONT>                                "some dimensions don't match: {0} != {1}",<a name="line.697"></a>
<FONT color="green">698</FONT>                                0, dimensionSize.length);<a name="line.698"></a>
<FONT color="green">699</FONT>                    }<a name="line.699"></a>
<FONT color="green">700</FONT>                    return null;<a name="line.700"></a>
<FONT color="green">701</FONT>                }<a name="line.701"></a>
<FONT color="green">702</FONT>                if (vector.length != dimensionSize.length) {<a name="line.702"></a>
<FONT color="green">703</FONT>                    throw MathRuntimeException.createIllegalArgumentException(<a name="line.703"></a>
<FONT color="green">704</FONT>                            "some dimensions don't match: {0} != {1}",<a name="line.704"></a>
<FONT color="green">705</FONT>                            vector.length,dimensionSize.length);<a name="line.705"></a>
<FONT color="green">706</FONT>                }<a name="line.706"></a>
<FONT color="green">707</FONT>    <a name="line.707"></a>
<FONT color="green">708</FONT>                Object[] lastDimension = (Object[]) multiDimensionalComplexArray;<a name="line.708"></a>
<FONT color="green">709</FONT>                for (int i = 0; i &lt; dimensionSize.length - 1; i++) {<a name="line.709"></a>
<FONT color="green">710</FONT>                    lastDimension = (Object[]) lastDimension[vector[i]];<a name="line.710"></a>
<FONT color="green">711</FONT>                }<a name="line.711"></a>
<FONT color="green">712</FONT>    <a name="line.712"></a>
<FONT color="green">713</FONT>                Complex lastValue = (Complex) lastDimension[vector[dimensionSize.length - 1]];<a name="line.713"></a>
<FONT color="green">714</FONT>                lastDimension[vector[dimensionSize.length - 1]] = magnitude;<a name="line.714"></a>
<FONT color="green">715</FONT>    <a name="line.715"></a>
<FONT color="green">716</FONT>                return lastValue;<a name="line.716"></a>
<FONT color="green">717</FONT>            }<a name="line.717"></a>
<FONT color="green">718</FONT>    <a name="line.718"></a>
<FONT color="green">719</FONT>            /**<a name="line.719"></a>
<FONT color="green">720</FONT>             * Get the size in all dimensions.<a name="line.720"></a>
<FONT color="green">721</FONT>             * @return size in all dimensions<a name="line.721"></a>
<FONT color="green">722</FONT>             */<a name="line.722"></a>
<FONT color="green">723</FONT>            public int[] getDimensionSizes() {<a name="line.723"></a>
<FONT color="green">724</FONT>                return dimensionSize.clone();<a name="line.724"></a>
<FONT color="green">725</FONT>            }<a name="line.725"></a>
<FONT color="green">726</FONT>    <a name="line.726"></a>
<FONT color="green">727</FONT>            /**<a name="line.727"></a>
<FONT color="green">728</FONT>             * Get the underlying storage array<a name="line.728"></a>
<FONT color="green">729</FONT>             * @return underlying storage array<a name="line.729"></a>
<FONT color="green">730</FONT>             */<a name="line.730"></a>
<FONT color="green">731</FONT>            public Object getArray() {<a name="line.731"></a>
<FONT color="green">732</FONT>                return multiDimensionalComplexArray;<a name="line.732"></a>
<FONT color="green">733</FONT>            }<a name="line.733"></a>
<FONT color="green">734</FONT>    <a name="line.734"></a>
<FONT color="green">735</FONT>            /** {@inheritDoc} */<a name="line.735"></a>
<FONT color="green">736</FONT>            @Override<a name="line.736"></a>
<FONT color="green">737</FONT>            public Object clone() {<a name="line.737"></a>
<FONT color="green">738</FONT>                MultiDimensionalComplexMatrix mdcm =<a name="line.738"></a>
<FONT color="green">739</FONT>                        new MultiDimensionalComplexMatrix(Array.newInstance(<a name="line.739"></a>
<FONT color="green">740</FONT>                        Complex.class, dimensionSize));<a name="line.740"></a>
<FONT color="green">741</FONT>                clone(mdcm);<a name="line.741"></a>
<FONT color="green">742</FONT>                return mdcm;<a name="line.742"></a>
<FONT color="green">743</FONT>            }<a name="line.743"></a>
<FONT color="green">744</FONT>            <a name="line.744"></a>
<FONT color="green">745</FONT>            /**<a name="line.745"></a>
<FONT color="green">746</FONT>             * Copy contents of current array into mdcm.<a name="line.746"></a>
<FONT color="green">747</FONT>             * @param mdcm array where to copy data<a name="line.747"></a>
<FONT color="green">748</FONT>             */<a name="line.748"></a>
<FONT color="green">749</FONT>            private void clone(MultiDimensionalComplexMatrix mdcm) {<a name="line.749"></a>
<FONT color="green">750</FONT>                int[] vector = new int[dimensionSize.length];<a name="line.750"></a>
<FONT color="green">751</FONT>                int size = 1;<a name="line.751"></a>
<FONT color="green">752</FONT>                for (int i = 0; i &lt; dimensionSize.length; i++) {<a name="line.752"></a>
<FONT color="green">753</FONT>                    size *= dimensionSize[i];<a name="line.753"></a>
<FONT color="green">754</FONT>                }<a name="line.754"></a>
<FONT color="green">755</FONT>                int[][] vectorList = new int[size][dimensionSize.length];<a name="line.755"></a>
<FONT color="green">756</FONT>                for (int[] nextVector: vectorList) {<a name="line.756"></a>
<FONT color="green">757</FONT>                    System.arraycopy(vector, 0, nextVector, 0,<a name="line.757"></a>
<FONT color="green">758</FONT>                                     dimensionSize.length);<a name="line.758"></a>
<FONT color="green">759</FONT>                    for (int i = 0; i &lt; dimensionSize.length; i++) {<a name="line.759"></a>
<FONT color="green">760</FONT>                        vector[i]++;<a name="line.760"></a>
<FONT color="green">761</FONT>                        if (vector[i] &lt; dimensionSize[i]) {<a name="line.761"></a>
<FONT color="green">762</FONT>                            break;<a name="line.762"></a>
<FONT color="green">763</FONT>                        } else {<a name="line.763"></a>
<FONT color="green">764</FONT>                            vector[i] = 0;<a name="line.764"></a>
<FONT color="green">765</FONT>                        }<a name="line.765"></a>
<FONT color="green">766</FONT>                    }<a name="line.766"></a>
<FONT color="green">767</FONT>                }<a name="line.767"></a>
<FONT color="green">768</FONT>                <a name="line.768"></a>
<FONT color="green">769</FONT>                for (int[] nextVector: vectorList) {<a name="line.769"></a>
<FONT color="green">770</FONT>                    mdcm.set(get(nextVector), nextVector);<a name="line.770"></a>
<FONT color="green">771</FONT>                }<a name="line.771"></a>
<FONT color="green">772</FONT>            }<a name="line.772"></a>
<FONT color="green">773</FONT>        }<a name="line.773"></a>
<FONT color="green">774</FONT>        <a name="line.774"></a>
<FONT color="green">775</FONT>        <a name="line.775"></a>
<FONT color="green">776</FONT>        /** Computes the n&lt;sup&gt;th&lt;/sup&gt; roots of unity. <a name="line.776"></a>
<FONT color="green">777</FONT>         * A cache of already computed values is maintained.<a name="line.777"></a>
<FONT color="green">778</FONT>         */<a name="line.778"></a>
<FONT color="green">779</FONT>        private static class RootsOfUnity implements Serializable {<a name="line.779"></a>
<FONT color="green">780</FONT>    <a name="line.780"></a>
<FONT color="green">781</FONT>          /** Serializable version id. */<a name="line.781"></a>
<FONT color="green">782</FONT>          private static final long serialVersionUID = 6404784357747329667L;<a name="line.782"></a>
<FONT color="green">783</FONT>    <a name="line.783"></a>
<FONT color="green">784</FONT>          /** Number of roots of unity. */<a name="line.784"></a>
<FONT color="green">785</FONT>          private int      omegaCount;<a name="line.785"></a>
<FONT color="green">786</FONT>    <a name="line.786"></a>
<FONT color="green">787</FONT>          /** Real part of the roots. */<a name="line.787"></a>
<FONT color="green">788</FONT>          private double[] omegaReal;<a name="line.788"></a>
<FONT color="green">789</FONT>    <a name="line.789"></a>
<FONT color="green">790</FONT>          /** Imaginary part of the roots for forward transform. */<a name="line.790"></a>
<FONT color="green">791</FONT>          private double[] omegaImaginaryForward;<a name="line.791"></a>
<FONT color="green">792</FONT>    <a name="line.792"></a>
<FONT color="green">793</FONT>          /** Imaginary part of the roots for reverse transform. */<a name="line.793"></a>
<FONT color="green">794</FONT>          private double[] omegaImaginaryInverse;<a name="line.794"></a>
<FONT color="green">795</FONT>    <a name="line.795"></a>
<FONT color="green">796</FONT>          /** Forward/reverse indicator. */<a name="line.796"></a>
<FONT color="green">797</FONT>          private boolean  isForward;<a name="line.797"></a>
<FONT color="green">798</FONT>    <a name="line.798"></a>
<FONT color="green">799</FONT>          /**<a name="line.799"></a>
<FONT color="green">800</FONT>           * Build an engine for computing then &lt;sup&gt;th&lt;/sup&gt; roots of unity<a name="line.800"></a>
<FONT color="green">801</FONT>           */<a name="line.801"></a>
<FONT color="green">802</FONT>          public RootsOfUnity() {<a name="line.802"></a>
<FONT color="green">803</FONT>            <a name="line.803"></a>
<FONT color="green">804</FONT>            omegaCount = 0;<a name="line.804"></a>
<FONT color="green">805</FONT>            omegaReal = null;<a name="line.805"></a>
<FONT color="green">806</FONT>            omegaImaginaryForward = null;<a name="line.806"></a>
<FONT color="green">807</FONT>            omegaImaginaryInverse = null;<a name="line.807"></a>
<FONT color="green">808</FONT>            isForward = true;<a name="line.808"></a>
<FONT color="green">809</FONT>            <a name="line.809"></a>
<FONT color="green">810</FONT>          }<a name="line.810"></a>
<FONT color="green">811</FONT>    <a name="line.811"></a>
<FONT color="green">812</FONT>          /**<a name="line.812"></a>
<FONT color="green">813</FONT>           * Check if computation has been done for forward or reverse transform.<a name="line.813"></a>
<FONT color="green">814</FONT>           * @return true if computation has been done for forward transform<a name="line.814"></a>
<FONT color="green">815</FONT>           * @throws IllegalStateException if no roots of unity have been computed yet<a name="line.815"></a>
<FONT color="green">816</FONT>           */<a name="line.816"></a>
<FONT color="green">817</FONT>          public synchronized boolean isForward() throws IllegalStateException {<a name="line.817"></a>
<FONT color="green">818</FONT>              <a name="line.818"></a>
<FONT color="green">819</FONT>            if (omegaCount == 0) {<a name="line.819"></a>
<FONT color="green">820</FONT>              throw MathRuntimeException.createIllegalStateException(<a name="line.820"></a>
<FONT color="green">821</FONT>                      "roots of unity have not been computed yet");<a name="line.821"></a>
<FONT color="green">822</FONT>            }        <a name="line.822"></a>
<FONT color="green">823</FONT>            return isForward;<a name="line.823"></a>
<FONT color="green">824</FONT>            <a name="line.824"></a>
<FONT color="green">825</FONT>          }<a name="line.825"></a>
<FONT color="green">826</FONT>          <a name="line.826"></a>
<FONT color="green">827</FONT>          /** Computes the n&lt;sup&gt;th&lt;/sup&gt; roots of unity.<a name="line.827"></a>
<FONT color="green">828</FONT>           * &lt;p&gt;The computed omega[] = { 1, w, w&lt;sup&gt;2&lt;/sup&gt;, ... w&lt;sup&gt;(n-1)&lt;/sup&gt; } where<a name="line.828"></a>
<FONT color="green">829</FONT>           * w = exp(-2 &amp;pi; i / n), i = &amp;sqrt;(-1).&lt;/p&gt;<a name="line.829"></a>
<FONT color="green">830</FONT>           * &lt;p&gt;Note that n is positive for<a name="line.830"></a>
<FONT color="green">831</FONT>           * forward transform and negative for inverse transform.&lt;/p&gt;<a name="line.831"></a>
<FONT color="green">832</FONT>           * @param n number of roots of unity to compute,<a name="line.832"></a>
<FONT color="green">833</FONT>           * positive for forward transform, negative for inverse transform<a name="line.833"></a>
<FONT color="green">834</FONT>           * @throws IllegalArgumentException if n = 0<a name="line.834"></a>
<FONT color="green">835</FONT>           */<a name="line.835"></a>
<FONT color="green">836</FONT>          public synchronized void computeOmega(int n) throws IllegalArgumentException {<a name="line.836"></a>
<FONT color="green">837</FONT>    <a name="line.837"></a>
<FONT color="green">838</FONT>            if (n == 0) {<a name="line.838"></a>
<FONT color="green">839</FONT>              throw MathRuntimeException.createIllegalArgumentException(<a name="line.839"></a>
<FONT color="green">840</FONT>                      "cannot compute 0-th root of unity, indefinite result");<a name="line.840"></a>
<FONT color="green">841</FONT>            }<a name="line.841"></a>
<FONT color="green">842</FONT>    <a name="line.842"></a>
<FONT color="green">843</FONT>            isForward = (n &gt; 0);<a name="line.843"></a>
<FONT color="green">844</FONT>            <a name="line.844"></a>
<FONT color="green">845</FONT>            // avoid repetitive calculations<a name="line.845"></a>
<FONT color="green">846</FONT>            final int absN = Math.abs(n);<a name="line.846"></a>
<FONT color="green">847</FONT>            <a name="line.847"></a>
<FONT color="green">848</FONT>            if (absN == omegaCount) {<a name="line.848"></a>
<FONT color="green">849</FONT>                return;<a name="line.849"></a>
<FONT color="green">850</FONT>            }<a name="line.850"></a>
<FONT color="green">851</FONT>    <a name="line.851"></a>
<FONT color="green">852</FONT>            // calculate everything from scratch, for both forward and inverse versions<a name="line.852"></a>
<FONT color="green">853</FONT>            final double t    = 2.0 * Math.PI / absN;<a name="line.853"></a>
<FONT color="green">854</FONT>            final double cosT = Math.cos(t);<a name="line.854"></a>
<FONT color="green">855</FONT>            final double sinT = Math.sin(t);<a name="line.855"></a>
<FONT color="green">856</FONT>            omegaReal             = new double[absN];<a name="line.856"></a>
<FONT color="green">857</FONT>            omegaImaginaryForward = new double[absN];<a name="line.857"></a>
<FONT color="green">858</FONT>            omegaImaginaryInverse = new double[absN];<a name="line.858"></a>
<FONT color="green">859</FONT>            omegaReal[0]             = 1.0;<a name="line.859"></a>
<FONT color="green">860</FONT>            omegaImaginaryForward[0] = 0.0;<a name="line.860"></a>
<FONT color="green">861</FONT>            omegaImaginaryInverse[0] = 0.0;<a name="line.861"></a>
<FONT color="green">862</FONT>            for (int i = 1; i &lt; absN; i++) {<a name="line.862"></a>
<FONT color="green">863</FONT>              omegaReal[i] =<a name="line.863"></a>
<FONT color="green">864</FONT>                omegaReal[i-1] * cosT + omegaImaginaryForward[i-1] * sinT;<a name="line.864"></a>
<FONT color="green">865</FONT>              omegaImaginaryForward[i] =<a name="line.865"></a>
<FONT color="green">866</FONT>                 omegaImaginaryForward[i-1] * cosT - omegaReal[i-1] * sinT;<a name="line.866"></a>
<FONT color="green">867</FONT>              omegaImaginaryInverse[i] = -omegaImaginaryForward[i];<a name="line.867"></a>
<FONT color="green">868</FONT>            }<a name="line.868"></a>
<FONT color="green">869</FONT>            omegaCount = absN;<a name="line.869"></a>
<FONT color="green">870</FONT>    <a name="line.870"></a>
<FONT color="green">871</FONT>          }<a name="line.871"></a>
<FONT color="green">872</FONT>    <a name="line.872"></a>
<FONT color="green">873</FONT>          /**<a name="line.873"></a>
<FONT color="green">874</FONT>           * Get the real part of the k&lt;sup&gt;th&lt;/sup&gt; n&lt;sup&gt;th&lt;/sup&gt; root of unity<a name="line.874"></a>
<FONT color="green">875</FONT>           * @param k index of the n&lt;sup&gt;th&lt;/sup&gt; root of unity<a name="line.875"></a>
<FONT color="green">876</FONT>           * @return real part of the k&lt;sup&gt;th&lt;/sup&gt; n&lt;sup&gt;th&lt;/sup&gt; root of unity<a name="line.876"></a>
<FONT color="green">877</FONT>           * @throws IllegalStateException if no roots of unity have been computed yet<a name="line.877"></a>
<FONT color="green">878</FONT>           * @throws IllegalArgumentException if k is out of range<a name="line.878"></a>
<FONT color="green">879</FONT>           */<a name="line.879"></a>
<FONT color="green">880</FONT>          public synchronized double getOmegaReal(int k)<a name="line.880"></a>
<FONT color="green">881</FONT>            throws IllegalStateException, IllegalArgumentException {<a name="line.881"></a>
<FONT color="green">882</FONT>            <a name="line.882"></a>
<FONT color="green">883</FONT>            if (omegaCount == 0) {<a name="line.883"></a>
<FONT color="green">884</FONT>                throw MathRuntimeException.createIllegalStateException(<a name="line.884"></a>
<FONT color="green">885</FONT>                        "roots of unity have not been computed yet");<a name="line.885"></a>
<FONT color="green">886</FONT>            }<a name="line.886"></a>
<FONT color="green">887</FONT>            if ((k &lt; 0) || (k &gt;= omegaCount)) {<a name="line.887"></a>
<FONT color="green">888</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.888"></a>
<FONT color="green">889</FONT>                        "out of range root of unity index {0} (must be in [{1};{2}])",<a name="line.889"></a>
<FONT color="green">890</FONT>                        k, 0, omegaCount - 1);<a name="line.890"></a>
<FONT color="green">891</FONT>            }<a name="line.891"></a>
<FONT color="green">892</FONT>            <a name="line.892"></a>
<FONT color="green">893</FONT>            return omegaReal[k];<a name="line.893"></a>
<FONT color="green">894</FONT>            <a name="line.894"></a>
<FONT color="green">895</FONT>          }<a name="line.895"></a>
<FONT color="green">896</FONT>    <a name="line.896"></a>
<FONT color="green">897</FONT>          /**<a name="line.897"></a>
<FONT color="green">898</FONT>           * Get the imaginary part of the k&lt;sup&gt;th&lt;/sup&gt; n&lt;sup&gt;th&lt;/sup&gt; root of unity<a name="line.898"></a>
<FONT color="green">899</FONT>           * @param k index of the n&lt;sup&gt;th&lt;/sup&gt; root of unity<a name="line.899"></a>
<FONT color="green">900</FONT>           * @return imaginary part of the k&lt;sup&gt;th&lt;/sup&gt; n&lt;sup&gt;th&lt;/sup&gt; root of unity<a name="line.900"></a>
<FONT color="green">901</FONT>           * @throws IllegalStateException if no roots of unity have been computed yet<a name="line.901"></a>
<FONT color="green">902</FONT>           * @throws IllegalArgumentException if k is out of range<a name="line.902"></a>
<FONT color="green">903</FONT>           */<a name="line.903"></a>
<FONT color="green">904</FONT>          public synchronized double getOmegaImaginary(int k)<a name="line.904"></a>
<FONT color="green">905</FONT>            throws IllegalStateException, IllegalArgumentException {<a name="line.905"></a>
<FONT color="green">906</FONT>          <a name="line.906"></a>
<FONT color="green">907</FONT>            if (omegaCount == 0) {<a name="line.907"></a>
<FONT color="green">908</FONT>                throw MathRuntimeException.createIllegalStateException(<a name="line.908"></a>
<FONT color="green">909</FONT>                        "roots of unity have not been computed yet");<a name="line.909"></a>
<FONT color="green">910</FONT>            }<a name="line.910"></a>
<FONT color="green">911</FONT>            if ((k &lt; 0) || (k &gt;= omegaCount)) {<a name="line.911"></a>
<FONT color="green">912</FONT>              throw MathRuntimeException.createIllegalArgumentException(<a name="line.912"></a>
<FONT color="green">913</FONT>                      "out of range root of unity index {0} (must be in [{1};{2}])",<a name="line.913"></a>
<FONT color="green">914</FONT>                      k, 0, omegaCount - 1);<a name="line.914"></a>
<FONT color="green">915</FONT>            }<a name="line.915"></a>
<FONT color="green">916</FONT>    <a name="line.916"></a>
<FONT color="green">917</FONT>            return (isForward) ?<a name="line.917"></a>
<FONT color="green">918</FONT>                omegaImaginaryForward[k] : omegaImaginaryInverse[k];<a name="line.918"></a>
<FONT color="green">919</FONT>            <a name="line.919"></a>
<FONT color="green">920</FONT>          }<a name="line.920"></a>
<FONT color="green">921</FONT>    <a name="line.921"></a>
<FONT color="green">922</FONT>        }<a name="line.922"></a>
<FONT color="green">923</FONT>        <a name="line.923"></a>
<FONT color="green">924</FONT>    }<a name="line.924"></a>




























































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